Get rich quick. This idea has enabled casinos and lotteries to make millions, and sometimes billions, of dollars each year. Many people gamble in hopes of winning some money. Realistically, the odds of winning a substantial amount of money without losing a significant amount are very low. The reason that casinos and lotteries make so much money is because the house (the casino or lottery) has an advantage. Probability is in their favor. And, in most cases, math can describe this considerable disadvantage to players.

Independent vs Dependent Events

People often misunderstand the notion of independent events. This is a probability term meaning that past events have no influence on future outcomes. For example, when flipping a coin four consecutive times, the probability of getting four heads is:

This is because the probability of flipping a head if you flip a coin once is ½. Flipping a coin is an example of an independent event. When flipping a coin, the probability of getting a head does not change no matter how many times you flip the coin. When the coin is flipped and the first three flips are heads, the fourth flip still has the probability of ½ However, many people misunderstand that the first three flips somehow influence the fourth flip, but they do not. The probability is still the same, as if the first three flips had never occurred.

This is a common misunderstanding when people are picking numbers for a lottery. Picking the same numbers every time does not guarantee that they will be picked eventually. Also, if a person picked 36 as one of their numbers and one of the drawn numbers was 35, it does not really imply that the person was close to winning. Although 35 and 36 are only one number apart, it does not mean that after 35 was picked, 36 would be the next number. In fact, the person is no closer than if 3 or 97 were picked. People often think this way, but it is just a misconception of independent events.

Gambling card games are not necessarily independent events. If the cards are not replaced into the deck, then probabilities change depending on which cards have been dealt. For example, the probability of being dealt an ace from a standard deck of 52 cards is 4/52 or 1/13. However, if the first person is dealt an ace, the probability that the second person will also be dealt an ace is now 3/51, if the first ace is not replaced into the deck. This applies to all card games, particularly Poker.

There are variations to the game of Poker, but the ranks of Poker hands are the same. The best Poker hand is a royal flush, which consists of a 10, jack, queen, king, and ace all of the same suit. This is considered the best hand because players have the lowest probability of being dealt this hand. The probability of being dealt a royal flush (RF), in a particular suit, say spades for example, in a five card hand can be explained as follows. The first card must be one of the five spades of 10, jack, queen, king, or ace, and hence has probability 5/52. The second card must be one of the remaining four top spades from the remaining 51 cards and hence has probability 4/51. The third card must be one of the remaining three top spades left from the remaining 50 cards and has a probability of 3/50. Likewise, the last two cards have probabilities of 2/49 and 1/48. Then the probability of being dealt a royal flush in spades in a five card hand is

Hence the probability of being dealt a royal flush in any suit in a five card hand is

In certain games, experienced gamblers may use a card counting system to aid them in their betting decisions. Card counting is commonly used when playing Blackjack. Blackjack is a card game where the player and the dealer both try to achieve a card count as close to 21 as possible, without going over 21. Card counting is a way for players to keep track of cards that have been dealt, giving them an idea about which cards are still left in the deck. To do this, they assign each card a numerical value. Card counting systems can vary, but basically if a low card is dealt, the player adds one, if a high card is dealt, they subtract one. A positive sum suggests that the player may win because if a lot of low cards have been dealt, there is a better chance that the dealer will bust (over 21) with a high card. Blackjack is one of few casino games where card counting can be useful because probabilities change as cards are dealt and the player can see many of the cards that are dealt.

Roulette

Unlike Blackjack, Roulette is a game involving independent events. When playing Roulette, a large wheel is spun and a small ball is dropped onto it. The ball will eventually fall into one of 38 spaces. The spaces are alternately colored black and red and numbered 1 – 36, with two additional green spaces marked with 0 and 00. Players can bet on numbers, combinations, ranges, odd/even, and colors.

Payouts with Ratios

The payout for a single number bet is 35:1. This is a ratio meaning that if a person bets one dollar on a 5 and the ball indeed lands in that 5 space, then the player will receive their one dollar back plus $35 for winning. So, for every dollar a person bets, they receive 35 times that much by winning on a numbered space, plus their original bet back. However, this ratio is not granted for every bet. If a person is betting $5 on red and they win, the payout is only 1:1, meaning that the person gets their $5 back and also wins $5. These different ratios are based on the probabilities of certain bets. For example, the probability of the ball landing in a 5 space is 1/38, whereas the probability of the ball landing in a red space is 18/38. This is why a numbered space has a 35:1 payout and a colored space is only 1:1.

It is also worth noting that betting on red is one of the highest probability bets in Roulette. If the probability of the ball landing in a red space is 18/38, then the probability of the ball not landing in a red space is:

This is the probability that the dealer or house has of winning, meaning that the house has a greater probability of winning than the player. This is how casinos make money.

Expected Value

Some games are considered more risky than others based on their probabilities of winning and their expected value. The expected value is what the player can expect to win or lose if they were to play many times with the same bet. For example, when playing Roulette, let’s say that a player bets $10 on red, with a payout of 1:1. The expected value of that bet played over and over can be expressed as follows.

The winning amount for one bet is $10

The losing amount for one bet is -$10

Expected value is calculated as follows:

[(probability of winning)(amount won per bet) - (probability of losing)(amount lost per bet)]

We can represent this mathematically using the values from above:

This means that if a player were to make this same bet of $10 on red over and over again, the player can expect to lose $0.53 for each bet of $10. A player has better chances of winning money with a positive expected value.

It is important for gamblers to understand what probabilities and odds mean. Consider this statement: your odds of winning a particular prize in a raffle are 1 in 10. This means that if only 10 tickets were sold and thrown into a bin, you would have one ticket. However, if the raffle sold 100 tickets and drew for 10 prizes, you are not guaranteed to win, even though the odds suggest you will. Odds can sometimes be misleading, so it is important to understand the math behind probabilities. Probabilities do not guarantee winnings; they simply propose a likelihood of winning.

Gambling is a popular source of entertainment for many people. By understanding the math behind gambling and betting games, people realize that their odds of winning are less than the odds of the house winning. Short term, players and the house have an almost equal opportunity of winning money, but in the long run, the house will win (expected values are negative for the player and the probabilities favor the house). This is fairly easy to see since casinos and lotteries are making an enormous amount of money with their business, all because they understand the mathematics of probability and winning

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.